Supposing the strait line B V (in fig. 1) be equal to the arch of the quadrant B H D, and A V be connected cutting the arch B H D in I, it may be asked what proportion the arch B I has to the arch I D. Let therefore the arch A Y be divided in the midst in _o_, and in the strait line A D let A _p_ be taken equal, and A _q_ double to the drawn chord A _o_. Then upon the centre A, with the radius A _q_, let an arch of a circle be drawn cutting the arch A Y in _r_, and let the arch Y _r_ be doubled at _t_; which being done, the drawn strait line A _t_ (by what has been last demonstrated) will be equal to the arch A Y. Again, upon the centre A with the radius A _t_ let the arch _t u_ be drawn cutting A D in _u_; and the strait line A _u_ will be equal to the arch A Y. From the point _u_ let the strait line _u s_ be drawn equal and parallel to the strait line A B, cutting M N in _x_, and bisected by M N in the same point _x_. Therefore the strait line A _x_ being drawn and produced till it meet with B C produced in V, it will cut off B V double to B _s_, that is, equal to the arch B H D. Now let the point, where the strait line A V cuts the arch B H D, be I; and let the arch D I be divided in the midst in _y_; and in the strait line D C, let D _z_ be taken equal, and D δ double to the drawn chord D _y_; and upon the centre D with the radius D δ let an arch of a circle be drawn cutting the arch B H D in the point _n_; and let the arch _n m_ be taken equal to the arch I _n_; which being done, the strait line D _m_ will (by the last foregoing corollary) be equal to the arch D I. If now the strait lines D _m_ and C V be equal, the arch B I will be equal to the radius A B or B C; and consequently X C being drawn, will pass through the point I. Moreover, if the semicircle B H D ϐ being completed, the strait lines ϐ I and B I be drawn, making a right angle (in the semicircle) at I, and the arch B I be divided in the midst at _i_, it will follow that A _i_ being connected will be parallel to the strait line ϐ I, and being produced to B C in _k_, will cut off the strait line B _k_ equal to the strait line _k_ I, and equal also to the strait line A γ cut off in A D by the strait line ϐ I. All which is manifest, supposing the arch B I and the radius B C to be equal.